§2.7 Derivatives of functions given implicitly

Introduction

Implicit Differentiation

Summary

Let $f$ be a differentiable function with no known formula. The derivative $\frac{d}{dx}[x^2 + f(x)]$ equals

The curve $x = y^5 - 5y^3 + 4y$ has $\frac{dy}{dx} = \frac{1}{5y^4 - 15y^2 + 4}$. Which statement about this formula is correct?

For the curve $x = y^5 - 5y^3 + 4y$, it is impossible to express $y$ as an explicit function of $x$ because the graph fails the vertical line test.

For the curve $x = y^5 - 5y^3 + 4y$, the tangent line at the point $(0, 1)$ is

For $x = y^5 - 5y^3 + 4y$, compute $\frac{dy}{dx}$ at $(0, 1)$ as a fraction.

For the same curve, vertical tangent lines occur exactly where $5y^4 - 15y^2 + 4 = 0$, which happens at $y \approx \pm 0.5439$ and $y \approx \pm 1.6444$. How many such points are there?

For $\sin(y) + y = x^3 + x$ at the point $(0,0)$, the tangent line is

The curve $y(y^2-1)(y-2) = x(x-1)(x-2)$ passes through $(1, 1)$, and implicit differentiation gives $\frac{dy}{dx} = \frac{(x-1)(x-2) + x(x-2) + x(x-1)}{(y^2-1)(y-2) + 2y^2(y-2) + y(y^2-1)}$. At $(1,1)$ this slope is

For an implicit curve whose slope formula is a single fraction in $x$ and $y$, the tangent is horizontal where

For the curve $\sin(x+y) + \cos(x-y) = 1$, implicit differentiation gives $\frac{dy}{dx} = \frac{\sin(x-y) - \cos(x+y)}{\cos(x+y) + \sin(x-y)}$. At $(\frac{\pi}{2}, \frac{\pi}{2})$ the tangent line is

For the curve $3xe^{-xy} = y^2$ at the point $(0.619061, 1)$, implicit differentiation gives $\frac{dy}{dx} = \frac{3e^{-xy} - 3xye^{-xy}}{3x^2e^{-xy}+2y}$. Compute the slope, rounded to three decimals.

For $x^3 - y^3 = 6xy$, compute $\frac{dy}{dx}$ at the point $(-3, 3)$.

Implicit differentiation proceeds by treating $y$ as a function of $x$ and differentiating both sides of the equation with respect to $x$.

Let $y = a^x$. Taking logarithms gives $x = \frac{\ln(y)}{\ln(a)}$, and differentiating both sides with respect to $x$ produces $1 = \frac{1}{\ln(a)} \cdot \frac{1}{y}\frac{dy}{dx}$. Solving yields